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A046192 Octagonal hexagonal numbers. 3

%I #34 Sep 08 2022 08:44:56

%S 1,11781,113123361,1086210502741,10429793134197921,

%T 100146872588357936901,961610260163619775927681,

%U 9233381617944204500099658261,88658929333889991446337142696641,851303030230630079923524744073490821,8174211607615580693535693146256516168801,78488779005021775588699645666830324179338581

%N Octagonal hexagonal numbers.

%C As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (sqrt(3) + sqrt(2))^8 = 4801 + 1960*sqrt(6). - _Ant King_, Dec 27 2011

%C Intersection of A000384 and A000567. - _Michel Marcus_, Jun 20 2015

%H Herman Jamke, <a href="/A046192/b046192.txt">Table of n, a(n) for n = 1..20</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalHexagonalNumber.html">Octagonal Hexagonal Number</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9603,-9603,1).

%F From _Ant King_, Dec 27 2011: (Start)

%F G.f.: x*(1 + 2178*x + 21*x^2)/((1-x)*(1 - 9602*x + x^2)).

%F a(n) = 9603*a(n-1) - 9603*a(n-2) + a(n-3).

%F a(n) = 9602*a(n-1) - a(n-2) + 2200.

%F a(n) = 1/96*((3*sqrt(3) - sqrt(2))*(sqrt(3) + sqrt(2))^(8n-5)+ (3*sqrt(3) + sqrt(2))*(sqrt(3) - sqrt(2))^(8n-5) - 22).

%F a(n) = floor(1/96*(3*sqrt(3) - sqrt(2))*(sqrt(3) + sqrt(2))^(8n-5)).

%F (End)

%p a:=5+2*sqrt(6): b:=5-2*sqrt(6): s:=n->a^n+b^n: d:=n->sqrt(6)*(a^n-b^n):for n from 0 to 40 do x:=simplify(s(n)-1/4*d(n)): y:=simplify(1/3*d(n)-s(n)/2): if(type((1+x/2)/3,integer) and type((1+y)/4,integer)) then printf("%d, ",(y^2-1)/8) fi: x:=simplify(s(n+1)+1/4*d(n+1)): y:=simplify(1/3*d(n+1)+s(n+1)/2): if(type((1+x/2)/3,integer) and type((1+y)/4,integer)) then printf("%d, ",(y^2-1)/8) fi: od: # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

%t LinearRecurrence[{9603, -9603, 1}, {1, 11781, 113123361}, 9] (* _Ant King_, Dec 27 2011 *)

%t CoefficientList[Series[(1 + 2178 x + 21 x^2) / ((1 - x) (1 - 9602 x + x^2)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 10 2017 *)

%o (Magma) I:=[1,11781,113123361]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Aug 10 2017

%Y Cf. A046190, A046191.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

%E More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

%E One more term from _Lior Manor_, Feb 13 2002

%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

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